The present invention relates to the field of electrical power generation and distribution systems.
Optimal power flow (OPF) algorithms based on successive linearization techniques are widely used to solve different problems in power system planning, operation and control.
Security Constrained OPF (SCOPF) problems are a special class of OPF problems which consider constraints derived from a normal system state (the xe2x80x9cbase casexe2x80x9d) and a set of predefined contingency states. SCOPF is an extension of the classical constrained economic dispatch problem in an effort to satisfy the system security requirements.
The definition of system security in actual power system operation varies throughout the power industry. Different operation policies and rules are applied to define security requirements. A widely accepted system security concept is so-called xe2x80x9cnxe2x88x921 security.xe2x80x9d Based on this concept, one of the main objectives in system operation and control is to keep the system in a normal state during normal system operation (the base case) and in the case of any one major contingency in the predefined list of contingencies. In order to satisfy the nxe2x88x921 security criteria the power system should be secure (no violations) after the occurrence of any single contingency in the system. This leads to the implementation of preventive control actions in the system, or the preventive mode of SCOPF.
SCOPF in preventive mode is conservative, because it does not consider the system""s post-contingency (corrective) control capabilities. By introducing corrective rescheduling to the nxe2x88x921 security concept, three different modes of control adjustments that affect the SCOPF solution can be identified: 1) Preventive mode; 2) Corrective mode; and 3) Preventive/corrective mode.
In the preventive mode, all control variables are optimized such that no post-contingency adjustments are necessary in order to avoid violation of base case and post-contingency constraints. This is the most secure solution mode, since no operator intervention is required following an anticipated contingency. The consequences of such a solution are a higher pre-contingency objective function, and a generally more difficult problem to solve. In some cases the preventive mode solution may not even exist, especially for more severe contingencies.
In the corrective mode, the control variables are permitted to adjust after the contingency occurs. This is a less secure mode of operation since operator action is required soon after the occurrence of a contingency to reach an acceptable operating state. Such a problem is generally easier to solve, since there are more degrees of freedom in the control adjustments. The corrective mode is solved as a sequence of independent optimization problems, one per contingency.
In the preventive/corrective mode, some of the violations for the violated constraints are relieved in the preventive mode, and the rest in the corrective mode. The preventive/corrective mode SCOPF produces a significantly larger optimization problem to be solved than the preventive mode SCOPF. However, it is more likely to have a feasible solution than the preventive mode SCOPF. It should be the preferred solution mode, especially in those cases where the preventive mode SCOPF requires expensive rescheduling of the base case generations. Normally operating a power system at a much higher cost in order to avoid limit violations in some contingency cases, may not be justifiable considering that the problem can be avoided by combining preventive and corrective control actions.
Execution of the SCOPF function in any of the previously mentioned modes is time consuming. Historically, the performance problems are dealt with by introducing in the model a relatively small number of critical contingencies. This approximation presents an unresolved modeling problem for all known SCOPF formulations. That is, by fixing just a small subset of most critical contingencies, there is no guaranty that other contingencies labeled as non-critical will not become critical after a new SCOPF solution. The only practical solution to this problem is to directly involve a large number of critical contingencies in the SCOPF formulation, resulting in a very large optimization problem to be solved.
Many different solution approaches have been proposed to solve the OPF problem. These methods can be generally classified into the following two categories: 1) successive linear programming (SLP) based methods; or 2) non-linear programming (NLP) based methods.
In the past, the SLP-based methods have been used almost exclusively for the solution of Security Constrained Economic Dispatch (SCED) problems. This is due to an inability of NLP-based methods to efficiently solve large numbers of cases simultaneously. An approach for the solution of the CED problem with piecewise linear cost curves and regulating margin constraints has been developed by Lugtu and Elacqua et al. (See R. Lugtu, xe2x80x9cSecurity Constrained Dispatchxe2x80x9d, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-98, pp. 270-274, January/February 1979; and A. J. Elacqua, et al., xe2x80x9cSecurity Constrained Dispatch at the New York Power Poolxe2x80x9d, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-101, pp. 2876-2883, August 1982.) That approach is based on the differential algorithm and the simplex method.
One approach has been to formulate the CED problem as a quadratic programming optimization problem and solved using Wolfe""s algorithm. (See G. F. Reid et al., xe2x80x9cEconomic Dispatch Using Quadratic Programmingxe2x80x9d, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-92, pp. 2015-2023, November/December 1973.) Furthermore, the Dantzig-Wolfe decomposition can be used efficiently for the solution of CED problems with reserve and contingency constraints. (See, e.g., M. Aganagic et al., xe2x80x9cSecurity Constrained Economic Dispatch Using Nonlinear Dantzig-Wolfe Decompositionxe2x80x9d, IEEE Transactions on Power Systems, Vol. PWRS-12, pp. 105-112, February 1997.)
An efficient NLP-based implementation is described in A. Monticelli et al., xe2x80x9cSecurity-Constrained Optimal Power Flow with Post-Contingency Corrective Reschedulingxe2x80x9d, IEEE Transactions on Power Systems, Vol. PWRS-2, pp. 175-182, February 1987. That approach is based on an AC power flow model and a generalized Benders decomposition. It is capable of solving SCOPF problems in both preventive and preventive/corrective modes.
Linear programming has been recognized as a reliable and robust technique for solving a large subset of specialized OPF problems characterized by linear separable objective functions and linear constraints. Many practical implementations of different OPF functions in modern energy management system (EMS) environments use an LP optimizer. Among various LP implementations probably the most efficient one is dual simplex successive linear programming with a special logic for traversing the segments of piecewise linearized cost curves called segment refinement. (See B. Stott et al., xe2x80x9cReview of Linear Programming Applied to Power System Reschedulingxe2x80x9d, IEEE PICA Conf. Proc., pp. 142-154, Cleveland, May 1979; Alsac et al., xe2x80x9cFurther Developments in LP-Based Optimal Power Flowxe2x80x9d, IEEE Transactions on Power Systems, Vol. PWRS-5, pp. 697-711, August 1990.) This method has been successfully implemented in solving general OPF problems including active loss minimization. It is very efficient in solving OPF problems with relatively small numbers of constraints and controls, which is not usually the case for SCOPF problems. The dual simplex LP algorithm and segment refinement are described in P. Ristanovic, xe2x80x9cSuccessive Linear Programming Based OPF Solutionxe2x80x9d, IEEE Tutorial Course, Optimal Power Flow: Solution Techniques, Requirements, and Challenges, 96 TP 111-0, 1996.
Interior point methods (IPMs) for mathematical programming problems were introduced by Frisch more than 30 years ago. Fiacco and McCormick further developed IPMs as a tool for the solution of nonlinear programming problems. (A. V. Fiacco et al., Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wile and Sons, New York, 1955). Interest in IPMs has increased since Karmarkar""s publication in 1984. (N. Karmarkar, xe2x80x9cA New Polynomial-Time Algorithm for Linear Programmingxe2x80x9d, Combinator, 1984). After Karmarkar""s publication it has been shown that his method is just a special case of general logarithmic barrier methods. This has further focused attention on the development of logarithmic barrier methods. Extensive research in this area in the last ten years has proven that IPMs are competitive with the simplex method in solving very large linear problems.
An early application of IPMs to power system optimization problems is reported in C. N. Lu et al., xe2x80x9cNetwork Constrained Security Control Using An Interior Point Algorithmxe2x80x9d, IEEE Transactions on Power Systems, Vol. PWRS-8, pp. 1068-1076, August 1993. The Constrained Economic Dispatch problem is solved using a successive linearization technique and a Primal Affine Scaling (PAS) interior point method. Test results on 6 to 118 bus systems verified good convergence and performance characteristics of PAS IPM. A Primal-Dual logarithmic Barrier algorithm has also been applied to general non-linear OPF problems.
It has been shown that Mehrotra""s predictor-corrector method can efficiently solve large-scale OPF problems with up to 2423 buses. (S. Mehrotra, xe2x80x9cOn the Implementation of A Primal-Dual Interior Point Methodxe2x80x9d, SIAM J. Optimization, Vol. 2, No. 3, 1992, pp. 575-601.) A similar formulation has been used to solve a large optimal reactive dispatch problem by a pure primal-dual logarithmic barrier method. (See S. Granville, xe2x80x9cOptimal Reactive Dispatch Through Interior Point Methodxe2x80x9d, IEEE Transactions on Power Systems, Vol. PWRS-9, pp. 136-146, February 1994.) A successive linear programming interior point method has been used to efficiently solve large SCED problems with up to 2124 buses.
The present invention provides a security-constrained OPF (SCOPF) process which employs a quadratic programming primal-dual interior point solution method. The present invention provides both preventive and preventive-corrective SCOPF.
The process of the present invention includes all of the favorable features of full non-linear optimization methods and successive linearization methods and has clear advantages over simplex-based OPF and SCOPF methods. An advantage of the process of the present invention is that its relative efficiency increases as the size of the optimization problem increases. This enables modeling of a very large number of contingencies and controls in SCOPF problems, allowing the solution of large preventive and preventive-corrective SCOPF problems in real-time. Another benefit is the accurate modeling of non-linear objective functions that may not be separable.
The process of the present invention provides for an efficient solution of highly non-linear OPF problems such as loss minimization, full optimization, voltage stability, and reactive SCOPF.
In an exemplary embodiment, the process of the present invention is efficiently used for bid evaluation and locational pricing and congestion management in deregulated energy markets. Other applications of the inventive process include OPF/SCOPF problems with discrete controls (LTCs, VCCs, VCRs, etc.) as well as security-constrained dynamic OPF (SCDOPF) problems.
A further exemplary embodiment of the present invention includes an infeasibility detection logic.